Maths-
General
Easy

Question

For x element of R Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x minus 3 over denominator x plus 2 end fraction close parentheses to the power of x equals

  1. e
  2. e to the power of negative 1 end exponent
  3. e to the power of negative 5 end exponent
  4. e to the power of 5

hintHint:

space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x minus 3 over denominator x plus 2 end fraction close parentheses to the power of x is form of bold 1 to the power of bold infinity bold space bold italic f bold italic o bold italic r bold italic m. This is known as an indeterminate form, because it is unknown. One to the power infinity is unknown because infinity itself is endless.

The correct answer is: e to the power of negative 5 end exponent


    x element of R space space space comma space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x minus 3 over denominator x plus 2 end fraction close parentheses to the power of x
    space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x minus 3 over denominator x plus 2 end fraction close parentheses to the power of x space equals space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator 1 minus begin display style 3 over x end style over denominator 1 plus begin display style 2 over x end style end fraction close parentheses to the power of x
space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator 1 minus begin display style 3 over x end style over denominator 1 plus begin display style 2 over x end style end fraction close parentheses to the power of x space equals space space space Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator 1 minus begin display style 3 over infinity end style over denominator 1 plus begin display style 2 over infinity end style end fraction close parentheses to the power of infinity space equals 1 to the power of infinity space space space space left parenthesis I n d e t e r m i n a t e space f o r m right parenthesis
    bold 1 to the power of bold infinity bold space bold italic f bold italic o bold italic r bold italic m this is known as an indeterminate form, because it is unknown. One to the power infinity is unknown because infinity itself is endless. Take a look at some examples of indeterminate forms. To solve this limit we will use the following formula -
    Error converting from MathML to accessible text.
    Error converting from MathML to accessible text.

    There are seven indeterminate forms which are typically considered in the literature
    {\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,~0^{0},~1^{\infty },{\text{ and }}\infty ^{0}.}:

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